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Middle East Technical University

EE 604 TERM PROJECT

Localization Using TDOA and FDOA

Project Report

Student:Ufuk Tamer

1674597

Instructor:Prof. Temel Engin

Tuncer

Date: 9/6/2015

EE 604 TERM PROJECT UFUK TAMER

pg. 1 LOCALIZATION USING TDOA AND FDOA

Table of Contents

Introduction ...................................................................................................................................... 2

1) TDOA Localization .................................................................................................................. 3

1.1 Addition of a New Variable ............................................................................................... 5

1.2 Taylor Series Expansion .................................................................................................... 5

2) FDOA Localization ................................................................................................................... 7

2.1 Nonlinear Least Squares Solution ..................................................................................... 8

2.2 Linear Least Squares Estimate .......................................................................................... 9

3) TDOA/FDOA Localization .................................................................................................... 10

3.1 Nonlinear Least Squares Solution ........................................................................................ 11

3.2 Linear Least Squares Estimate ............................................................................................. 11

4) Simulation Results .................................................................................................................. 13

4.1 TDOA Simulation ................................................................................................................ 13

4.2 FDOA Simulation ................................................................................................................ 16

4.3 TDOA/FDOA Simulation .................................................................................................... 17

Conclusion ...................................................................................................................................... 19



Introduction

A wide range of methods are currently being used to geolocate the signals. The fundamental

techniques among them include Triangulation, TOA(Time of Arrival), TDOA(Time Difference of

Arrival), FDOA(Frequency Difference of Arrival) and hybrid techniques (e.g. joint

TDOA/FDOA). Triangulation uses DOA(Direction of Arrival) angle estimates and intersect the

line of bearings for position fixing. In TOA, time of flight from source to sensor is known and

target location is estimated based on TOA equations. This method requires synchronization with

transmitter. However, in TDOA method difference of arrival times are used to estimate the target

position and hence there is no need to synchronize with transmitter but sensors should be

synchronized to each other. In FDOA, Doppler frequency shifts are used for source localization.

In this project TDOA, FDOA and joint TDOA/FDOA methods for three-dimensional localization

are implemented in MATLAB.

In the first part, TDOA method is presented and the location estimation problem is solved for both

linearizing the equations and using Taylor expansion and approximation. In the second part, FDOA

method is discussed and equations are solved by using the similarities with TDOA. The third part

of this report shows the joint TDOA/FDOA location estimation method. Lastly, simulation results

are presented for all mentioned methods with some comparisons and conclusions based on the

performance of them.



1) TDOA Localization

Time difference of arrival localization method has been widely used for different applications. In

the civil technology, it is used in mobile communication systems to perform location of cell phones

of their subscribers. Also, boats and ship’s location is estimated by using acoustic waves TDOA

information. In the military technology, it is used to locate the enemy’s emitting device such as

communication device in the battle field.

The advantages of TDOA method can be written as,

In triangulation an array is used for each measurement and at least 2 DF(direction finding)

sites are required. In TDOA a single antenna per sensor and at least 4 sensors are required

for 3-D location estimation.

Usually higher precision and accuracy can be obtained.

The disadvantage of using TDOA is that accurate and synchronized clocks are needed for each

sensor. And TDOA estimation accuracy depends on,

Measurement errors on sensor positions

Multipath problem

Timing accuracy of sensors

Sensor geometry with respect to target

TDOA localization scenario is shown in Figure 1. Each sensor receives the signal with some delay

in time and frequency. Received signal for 𝑖th sensor is shown as,

𝑦𝑖(𝑡) = 𝑒𝑗𝛼𝑖𝑒−𝑗𝑤𝑑𝑖𝑡𝑠(𝑡 − 𝜏𝑖) + 𝑛(𝑡), 𝑖 = 1,2, … ,𝑀

where 𝛼𝑖 is the phase introduced by time of flight of the signal and 𝑤𝑑𝑖 is the Doppler frequency

shift of 𝑖th sensor with velocity 𝑣𝑖

𝛼𝑖 = 𝑤𝑐𝜏𝑖 , 𝑤𝑑𝑖=

𝑤𝑐𝑣𝑖

𝑐



Figure 1: TDOA Scenario

In TDOA scenario, sensors are assumed to be stationary and synchronized to each other. Hence

Doppler shifts are assumed to be zero for all sensors. The classical approach for estimating TDOA

is to compute the cross correlation between signals arriving different sensors. Hence,

𝑅𝑦1𝑦2(𝜏12) = ∫ 𝑦1(𝑡)𝑦2

∗(𝑡 − 𝜏12)𝑑𝑡𝑇

0

finding the peak of |𝑅𝑦1𝑦2(𝜏12)| gives TDOA value between 1st and 2nd sensors.

𝜏12 = 𝜏2 − 𝜏1

we can write 𝜏12 by using the 𝑑1 and 𝑑2 as,

𝜏12 =𝑑2 − 𝑑1

𝑐

and

𝑑2 = 𝑑1 + 𝑐𝜏12

𝑑𝑖 = ||𝒑𝑖 − 𝒑𝑇||

𝑑𝑖 = √(𝑥𝑖 − 𝑥𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2, 𝑖 = 1,2, … ,𝑀

𝑑𝑖2 = (𝑑1 + 𝑐𝜏1𝑖)

2 = (𝑥𝑖 − 𝑥𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2, 𝑖 = 2,3, … ,𝑀 (∗)

The solution of (∗) can be done with two different approaches,

1. Addition of a new variable and linearization (at least 5 sensors)

2. Taylor series expansion (at least 4 sensors)



1.1 Addition of a New Variable

The distance of 1st sensor equation is taken as reference and it is subtracted from the equation of

𝑖th sensor. Hence the equations are formed as,

𝑑12 = (𝑥1 − 𝑥𝑇)2 + (𝑦1 − 𝑦𝑇)2 + (𝑦1 − 𝑦𝑇)2

𝑑𝑖2 = (𝑑1 + 𝑐𝜏1𝑖)

2 = (𝑥𝑖 − 𝑥𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2

𝑑12 is subtracted from 𝑑𝑖

2 and for 𝑖 = 2

2𝑑1𝑐𝜏12 + 𝑐2𝜏122 = 𝑥2

2 + 𝑦22 + 𝑧2

2 − 𝑥12 − 𝑦1

2 − 𝑧12 − 2(𝑥2 − 𝑥1)𝑥𝑇 − 2(𝑦2 − 𝑦1)𝑦𝑇 − 2(𝑧2 − 𝑧1)𝑧𝑇

Equations are written in matrix form for 𝑖 = 2,3, … ,𝑀

−2 [

(𝑥2 − 𝑥1) (𝑦2− 𝑦

1) (𝑧2 − 𝑧1) 𝑐𝜏12

⋮ (𝑥𝑀 − 𝑥1) (𝑦

𝑀− 𝑦

1) (𝑧𝑀 − 𝑧1) 𝑐𝜏1𝑀

] [

𝑥𝑇

𝑦𝑇

𝑧𝑇

𝑑1

] =

[

𝑐2𝜏122 − 𝑥2

2 − 𝑦22 − 𝑧2

2 + 𝑥12 + 𝑦

12 + 𝑧1

2

𝑐2𝜏132 − 𝑥3

2 − 𝑦32 − 𝑧3

2 + 𝑥12 + 𝑦

12 + 𝑧1

2

⋮𝑐2𝜏1𝑀

2 − 𝑥𝑀2 − 𝑦

𝑀2 − 𝑧𝑀

2 + 𝑥12 + 𝑦

12 + 𝑧1

2]

𝑨𝒙 = 𝒃

�̂� = (𝑨𝑯𝑨)−1𝑨𝑯𝒃

3-D coordinates can be found if there are M>5. It is desired to have more sensors and an over

determined set of equations to improve the accuracy.

1.2 Taylor Series Expansion

The equation of 𝑑𝑖 can be linearized by a Taylor series expansion about an initial estimate of target

location 𝒑𝑇0= [𝑥𝑇0

𝑦𝑇0 𝑧𝑇0

]𝑇

𝑓(𝑥) = 𝑓(𝑥𝑇0) +

𝑓′(𝑥𝑇0)

1!(𝑥𝑇 − 𝑥𝑇0

) +𝑓′′(𝑥𝑇0

)

2!(𝑥𝑇 − 𝑥𝑇0

)2+ ⋯

Only the first order term is kept and the rest is ignored.

𝑃𝑖 = 𝑐𝜏1𝑖 = 𝑓𝑖(𝒑𝑇) = 𝑑𝑖 − 𝑑1 = ||𝒑𝑇 − 𝒑𝑖|| − ||𝒑𝑇 − 𝒑1||, 𝑖 = 2,3, … ,𝑀

𝑃𝑖’s are known and 𝑓𝑖(. )’s will be linearized by using Taylor series expansion with initial target

position,

𝑃𝑖 − 𝑓𝑖|𝒑𝑇0=

𝜕𝑓𝑖𝜕𝑥𝑇

|𝒑𝑇0(𝑥𝑇 − 𝑥𝑇0

) +𝜕𝑓𝑖𝜕𝑦𝑇

|𝒑𝑇0(𝑦𝑇 − 𝑦𝑇0

) +𝜕𝑓𝑖𝜕𝑧𝑇

|𝒑𝑇0(𝑧𝑇 − 𝑧𝑇0

)

Δ𝑥𝑇 = 𝑥𝑇 − 𝑥𝑇0



Δ𝑦𝑇 = 𝑦𝑇 − 𝑦𝑇0

Δ𝑧𝑇 = 𝑧𝑇 − 𝑧𝑇0

Equations are written in matrix form for 𝑖 = 2,3, … ,𝑀

[

𝑃2 − 𝑓2𝑃3 − 𝑓3

⋮𝑃𝑀 − 𝑓𝑀

] =

[ 𝜕𝑓2𝜕𝑥𝑇

𝜕𝑓2𝜕𝑦𝑇

𝜕𝑓2𝜕𝑧𝑇

𝜕𝑓3𝜕𝑥𝑇

𝜕𝑓3𝜕𝑦𝑇

𝜕𝑓3𝜕𝑧𝑇

⋮ 𝜕𝑓𝑀𝜕𝑥𝑇

𝜕𝑓𝑀𝜕𝑦𝑇

𝜕𝑓𝑀𝜕𝑧𝑇 ]

[Δ𝑥𝑇

Δ𝑦𝑇

Δ𝑧𝑇

]

𝑨𝚫𝒙 = 𝑷

3-D coordinates can be found if there are M>4 for this approach.

𝚫�̂� = (𝑨𝑯𝑨)−1𝑨𝑯𝑷

�̂� = 𝚫�̂� + 𝒑𝑇0



2) FDOA Localization

When there relative motions between receivers and an emitter, frequency difference of arrival

method can be used to estimate the emitter position. FDOA or differential Doppler is a technique

which is similar to TDOA, to estimate the location of an emitter by using frequency shifts between

receivers. Likewise, it can also be used to detect receiver position based on multiple emitters.

FDOA is especially used with UAVs (Unmanned Aerial Vehicles) and satellites to estimate the

target position in practical systems.

The accuracy of FDOA estimation depends on

Bandwidth of the signal

SNR at each observation point

Sensor geometry and vector velocities of sensors and emitter

FDOA localization scenario is shown in Figure 2. Each sensor receives the signal with some delay

in time and frequency. Received signal for 𝑖th sensor is shown as,





𝑤𝑐𝑣𝑖

𝑐

Figure 2: FDOA Scenario

In FDOA scenario, target is assumed to be stationary and sensors are in motion with known

velocities. Hence each sensor received signal has Doppler shift frequency and the classical

approach for estimating differential Doppler shifts is to compute the cross correlation in Fourier

domain between signals. So,



𝑅𝑦1𝑦2(𝑤𝑑12

) = ∫ 𝑦1(𝑡)𝑦2∗(𝑡)𝑒−𝑗𝑤𝑑12𝑑𝑡

𝑇

0

finding the peak of |𝑅𝑦1𝑦2(𝑤𝑑12

)| gives FDOA value (angular frequency) between 1st and 2nd

sensors.

𝑤𝑑12= 𝑤𝑑2

− 𝑤𝑑1

𝑓𝑑12=

𝑓𝑐(𝑣2 − 𝑣1)

𝑐

we can write the equation in a different way as,

𝑓𝑑12=

𝑓𝑐𝑐

𝑑(𝑑2 − 𝑑1)

𝑑𝑡

and,

𝑑𝑑𝑖

𝑑𝑡=

(𝒑𝑖 − 𝒑𝑇)𝑇𝒗𝑖

𝑑𝑖

𝒗𝑖 = [𝑣𝑥𝑖 𝑣𝑦𝑖

𝑣𝑧𝑖]𝑇

writing the FDOA equations in terms of velocities as,

𝑓𝑑1𝑖=

𝑓𝑐𝑐

[(𝒑𝑖 − 𝒑𝑇)𝑇𝒗𝑖

𝑑𝑖−

(𝒑1 − 𝒑𝑇)𝑇𝒗1

𝑑1] , 𝑖 = 2,3, … ,𝑀

and in open form,

𝑓𝑑1𝑖=

𝑓𝑐𝑐

[(𝑥𝑖 − 𝑥𝑇)𝑣𝑥𝑖

+ (𝑦𝑖 − 𝑦𝑇)𝑣𝑦𝑖+ (𝑧𝑖 − 𝑧𝑇)𝑣𝑧𝑖

√(𝑥𝑖 − 𝑥𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2 + (𝑦𝑖 − 𝑦𝑇)2−

(𝑥1 − 𝑥𝑇)𝑣𝑥1+ (𝑦1 − 𝑦𝑇)𝑣𝑦1

+ (𝑧1 − 𝑧𝑇)𝑣𝑧1

√(𝑥1 − 𝑥𝑇)2 + (𝑦1 − 𝑦𝑇)2 + (𝑦1 − 𝑦𝑇)2]

equations can be written for 𝑖 = 2,3,… ,𝑀. Hence we have 𝑀 − 1 nonlinear equations and target location

has to be estimated based on these. To solve these nonlinear equations we perform two different approaches

1. Solving with nonlinear least squares estimate

2. Linear least squares estimate assuming that we know (measure) 𝑑𝑖’s

2.1 Nonlinear Least Squares Solution

We have a set of nonlinear equations as,

𝐹𝑖(𝒑𝑇 , 𝒑𝑖 , 𝒗𝑖) = 𝑓𝑑1𝑖−

𝑓𝑐𝑐


𝑑𝑖−


𝑑1] = 0, 𝑖 = 2,3, …𝑀



Nonlinear least squares solution states that the sum of 𝐹𝑖(𝒑𝑇 , 𝒑𝑖 , 𝒗𝑖)’s is minimized over 𝒑𝑇. Hence

the estimate is,

�̂�𝑇 = argmin𝒑𝑇

∑𝐹𝑖(𝒑𝑇 , 𝒑𝑖, 𝒗𝑖)2

𝑀

𝑖=2

This equation is too complex to compute for nonlinear squares solution procedure. In the simulation

part, MATLAB’s fsolve function is used for simplicity. The function solves these nonlinear set

of equations in an iterative fashion with an initial estimate of target position. More details will be

given at simulation results.

2.2 Linear Least Squares Estimate

In this approach, 𝑑𝑖’s are assumed to be known by sensors. With this assumption, nonlinear set of

equations become,

𝑓𝑑1𝑖=

𝑓𝑐𝑐


𝑑𝑖−


𝑑1]

𝛼𝑖 =𝑓𝑑1𝑖

𝑐

𝑓𝑐𝑑𝑖𝑑1 = 𝑑1(𝒑𝑖 − 𝒑𝑇)𝑇𝒗𝑖 − 𝑑𝑖(𝒑1 − 𝒑𝑇)𝑇𝒗1

𝛽𝑖 = 𝛼𝑖 − 𝑑1𝒑𝑖𝑇𝒗𝑖 + 𝑑𝑖𝒑1

𝑇𝒗1 = 𝒑𝑇𝑇(𝑑𝑖𝒗1 − 𝑑1𝒗𝑖) , 𝑖 = 2,3, … ,𝑀

[

𝛽2

𝛽3

⋮𝛽𝑀

] =

[ 𝑑2𝑣𝑥1

− 𝑑1𝑣𝑥2𝑑2𝑣𝑦1

− 𝑑1𝑣𝑦2𝑑2𝑣𝑧1

− 𝑑1𝑣𝑧2

𝑑3𝑣𝑥1− 𝑑1𝑣𝑥3

𝑑3𝑣𝑦1− 𝑑1𝑣𝑦3

𝑑3𝑣𝑧1− 𝑑1𝑣𝑧3

⋮ 𝑑𝑀𝑣𝑥1

− 𝑑1𝑣𝑥𝑀𝑑𝑀𝑣𝑦1

− 𝑑1𝑣𝑦𝑀𝑑𝑀𝑣𝑧1

− 𝑑1𝑣𝑧𝑀]

[

𝑥𝑇

𝑦𝑇

𝑧𝑇

]

𝑨𝒙 = 𝜷

�̂� = (𝑨𝑯𝑨)−1𝑨𝑯𝜷

Linear least squares estimate is computed as shown above for 𝑀 ≥ 4 under the assumption of

knowing 𝑑𝑖’s. For higher M values, the estimate will be more consistent.



3) TDOA/FDOA Localization

In order to improve the location estimation accuracy, TDOA and FDOA methods are used together.

Joint TDOA/FDOA method is especially used in satellite systems for more accurate location

estimation procedure. Accuracy of target depends on the same factors as well as TDOA and FDOA.

TDOA/FDOA localization scenario is shown in Figure 3. Each sensor receives the signal with

some delay in time and frequency. Received signal for 𝑖th sensor is shown as,





𝑤𝑐𝑣𝑖

𝑐

Figure 3: TDOA/FDOA Scenario

In TDOA/FDOA scenario, target is assumed to be stationary and sensors are in motion with known

velocities. The classical approach for estimating TDOA and FDOA parameters is to compute the

cross correlation between received signals both in time and frequency domain. It is also called

ambiguity function.

𝐴𝑦1𝑦2(𝜏12, 𝑤𝑑12

) = ∫ 𝑦1(𝑡)𝑦2∗(𝑡 − 𝜏12)𝑒

−𝑗𝑤𝑑12𝑑𝑡𝑇

0

Finding the peak of |𝐴𝑦1𝑦2(𝜏12, 𝑤𝑑12

)| gives both TDOA (𝜏12) and FDOA (𝑤𝑑12). After computing

the parameters, the equations are formed as,

𝑑𝑖 = 𝑑1 + 𝑐𝜏1𝑖 , 𝑖 = 2,3, … ,𝑀 (1)



𝑓𝑑1𝑖=

𝑓𝑐𝑐


𝑑𝑖−


𝑑1] , 𝑖 = 2,3, … ,𝑀 (2)

Substitute (1) into (2),

𝑓𝑑1𝑖=

𝑓𝑐𝑐


𝑑1 + 𝑐𝜏1𝑖−


𝑑1] , 𝑖 = 2,3, … ,𝑀

Joint TDOA/FDOA equation is found. We can compute those by,

1. Using nonlinear least squares estimate

2. Assuming that we know 𝑑1 and use linear least squares estimate

3.1 Nonlinear Least Squares Solution

We have two different nonlinear set of equations. For FDOA,

𝐹𝑖(𝒑𝑇 , 𝒑𝑖 , 𝒗𝑖) = 𝑓𝑑1𝑖−

𝑓𝑐𝑐


𝑑𝑖−


𝑑1] = 0, 𝑖 = 2,3, …𝑀

and for TDOA,

𝐺𝑖(𝒑𝑇 , 𝒑𝑖) = 𝑑𝑖 − 𝑑1 − 𝑐𝜏1𝑖 = 0 , 𝑖 = 2,3, … ,𝑀

Nonlinear least squares solution states that the sum of 𝐹𝑖(𝒑𝑇 , 𝒑𝑖 , 𝒗𝑖)2 + 𝐺𝑖(𝒑𝑇 , 𝒑𝑖)

2’s is minimized

over 𝒑𝑇. Hence the estimate is

�̂�𝑇 = argmin𝒑𝑇

[∑𝐹𝑖(𝒑𝑇 , 𝒑𝑖, 𝒗𝑖)2

𝑀

𝑖=2

+ ∑𝐺𝑖(𝒑𝑇 , 𝒑𝑖)2

𝑀

𝑖=2

]

Same procedure is applied to solve nonlinear set of equations for joint TDOA/FDOA method as it

is applied for FDOA.

3.2 Linear Least Squares Estimate

Joint TDOA/FDOA equation can be linearized if we know the distance of sensor which it is taken

as reference for finding TDOA/FDOA values. Hence we have,

𝑓𝑑1𝑖=

𝑓𝑐𝑐


𝑑1 + 𝑐𝜏1𝑖−


𝑑1] , 𝑖 = 2,3, … ,𝑀

𝛼𝑖 =𝑓𝑑1𝑖

𝑐

𝑓𝑐



(𝑑12 + 𝑑1𝑐𝜏1𝑖)𝛼𝑖 − 𝑑1𝒑𝑖

𝑇𝒗𝑖 + 𝑑1𝒑1𝑇𝒗1 + 𝑐𝜏1𝑖𝒑1

𝑇𝒗1 = 𝒑𝑇𝑇(𝑐𝜏1𝑖𝒗1 + 𝑑1𝒗1 − 𝑑1𝒗𝑖), 𝑖 = 2,3, … ,𝑀

We write the equations in matrix form for 𝑖 = 2,3, … ,𝑀

[(𝑑1

2 + 𝑑1𝑐𝜏12)𝛼2 − 𝑑1𝒑2𝑇𝒗2 + 𝑑1𝒑1

𝑇𝒗1 + 𝑐𝜏12𝒑1𝑇𝒗1

⋮(𝑑1

2 + 𝑑1𝑐𝜏1𝑀)𝛼𝑀 − 𝑑1𝒑𝑀𝑇 𝒗𝑀 + 𝑑1𝒑1

𝑇𝒗1 + 𝑐𝜏1𝑀𝒑1𝑇𝒗1

] = [𝑥1(𝑐𝜏12 + 𝑑1) − 𝑑1𝑥2 𝑦1(𝑐𝜏12 + 𝑑1) − 𝑑1𝑦2 𝑦1(𝑐𝜏12 + 𝑑1) − 𝑑1𝑦2

⋮ 𝑥1(𝑐𝜏1𝑀 + 𝑑1) − 𝑑1𝑥𝑀 𝑦1(𝑐𝜏1𝑀 + 𝑑1) − 𝑑1𝑦𝑀 𝑦1(𝑐𝜏1𝑀 + 𝑑1) − 𝑑1𝑦𝑀

] [

𝑥𝑇

𝑦𝑇

𝑧𝑇

]

And we find least squares estimate as follows,

𝑨𝒙 = 𝒃

�̂� = (𝑨𝑯𝑨)−1𝑨𝑯𝒃

The solution is valid for 𝑀 ≥ 4 and for higher values estimate is getting close the true value.



4) Simulation Results

In order to see the performance of all methods, simulations are done in MATLAB. In the

simulations, the following parameters are used. Since Matlab code is written in parametric, we

observe the performance of all methods in various conditions.

Sensor positions ±𝟑𝟎𝒎 uniformly dist. for each dimension

Target position ±300𝑚 uniformly dist. for each dimension

measurement error(for TDOA and FDOA) Gaussian random variable ∼ 𝒩(0, 𝜎𝑒2)

Sensor velocities (for FDOA) ±50m/s uniformly dist. for each dimension

4.1 TDOA Simulation

TDOA simulations are done for both estimation approaches. In Figure 4, one experiment is

performed for both methods. As you see, 5 sensors are used and linear solution (addition of one

variable) estimates better the target location compared to Taylor expansion solution for this

experiment. In the simulations, initial estimate for Taylor expansion approach is

𝒑𝑇0= 𝒑𝑇 + 𝒏

used. Here, 𝒏 is i.i.d. Gaussian random vector with zero mean and 𝜎𝑛2 variance. We take 𝜎𝑛 = 40,

which means standard deviation of initial estimate is 40m.

Figure 4: TDOA Experiment in MATLAB 𝜎𝑒 = 0.1𝑛𝑠𝑒𝑐

-300-200

-1000

100200

300

-300

-200

-100

0

100

200

300-300

-200

-100

0

100

200

300

x-axisy-axis

z-a

xis

Sensors

Target

Linear Solution

Taylor Expansion Solution



Simulations are done for large number of experiments and distance errors are computed for each

method. In Figure 6 and Figure 5, you see that there is a huge estimation errors in some experiments

up to 150km. This error comes from measurement errors and matrix inverse operations. In order to

see the RMSE performance of each method, we first eliminate such big errors by using a threshold

named “fail_thr” and the ones which are above this threshold value are assumed to be “fails”. The

rest is computed for RMSE performance. In the simulations, “fail_thr” is taken as 1km. In Figure

7, RMSE performance of linear solution approach is shown for M=5 sensors.

Figure 7: RMSE Performance of Linear Solution with 𝑀 = 5 sensors and 104 Monte Carlo trials

In Figure 8, Taylor series approach is shown with M=4,

0 0.5 1 1.5 2 2.5 3

x 10-9

0

50

100

150

200

250

300

350

e (sec)

RM

SE

(m

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

4

Dis

tan

ce

Err

or

(m)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

5

10

15x 10

4

Experiments

Dis

tan

ce

Err

or

(m)

Figure 6: TDOA Linear Solution: M=5, 𝜎𝑒 = 0.1𝑛𝑠𝑒𝑐 Figure 5: TDOA Taylor Expansion Solution: M=5, 𝜎𝑒 = 0.1𝑛𝑠𝑒𝑐,

σn = 40m



Figure 8: RMSE Performance of Taylor Series Solution with 𝑀 = 4 and σn = 40m (104 Monte Carlo trials)

For the same number of sensors, both approaches are compared. Since an initial estimate is used

for Taylor expansion approach, this is not a perfect comparison. But it gives a rough idea. For the

same number of sensors, linear solution outperforms Taylor expansion for the whole range of error

deviation. Especially for smaller measurement errors, linear solution estimates much better.

Figure 9: RMSE Performance Comparison of TDOA Solution Approaches 𝑀 = 5 and σn = 40m (104 Monte Carlo trials)

We also run the simulations for higher number of sensors. In Figure 10, same comparison is done

for 𝑀 = 10. This time initial estimate for Taylor expansion approach made better (𝜎𝑛 = 20𝑚). As

0 0.5 1 1.5 2 2.5 3

x 10-9

280

300

320

340

360

380

400

420

e (sec)

RM

SE

(m

)

0 0.5 1 1.5 2 2.5 3

x 10-9

0

50

100

150

200

250

300

350

400

e (sec)

RM

SE

(m

)

Linear Solution

Taylor Expansion



you see, performance is better for low measurement error deviation. However, as it increases,

Taylor approach gives better results due to the better initial estimate.

Figure 10: RMSE Performance Comparison of TDOA Solution Approaches 𝑀 = 10 and σn = 20m (104 Monte Carlo trials)

4.2 FDOA Simulation

FDOA simulations are done only for nonlinear least squares solution. As it is mentioned in FDOA

section, the set of nonlinear equations are solved by using Matlab’s fsolve function. fsolve

is commonly used for nonlinear equations and it solves a problem specified by 𝐹(𝑥) = 0 for x,

where 𝐹(𝑥) is a function that returns a vector value. It uses “Levenberg–Marquardt algorithm” to

solve nonlinear equations in an iterative fashion. Sometimes, the solution is diverges to some value

which is very far away from the real target position so we use “fail_thr” for this part too.

In Figure 11, RMSE performance of FDOA localization is shown for M=4. Computational

complexity is much higher than TDOA solution approaches, but it gives really good estimates even

for M=4. For higher number of sensors, RMSE performance is really good. Also, initial estimate

error deviation is taken 30meter for this part.

0 0.5 1 1.5 2 2.5 3

x 10-9

0

50

100

150

200

250

e (sec)

RM

SE

(m

)

Linear Solution

Taylor Expansion



Figure 11: RMSE Performance of Nonlinear Solution with 𝑀 = 4 sensors, σn = 30m and 103 Monte Carlo trials

Figure 12: RMSE Performance of Nonlinear Solution with 𝑀 = 8 sensors, σn = 30m and 103 Monte Carlo trials

4.3 TDOA/FDOA Simulation

Only nonlinear least squares solution is implemented in MATLAB for joint TDOA/FDOA method.

Similarly, fsolve function is used to find the target position from a set of nonlinear equations.

Since we have TDOA and FDOA information, there will be 2(𝑀 − 1) equations and the estimate

is much better compared to FDOA method.

In Figure 13, RMSE performance of TDOA/FDOA is shown for M=3.

0 1 2 3 4 5 6 7 8 9 1020

40

60

80

100

120

140

160

180

200

RM

SE

(m

)

e (Hz)

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

RM

SE

(m

)

e (Hz)



Figure 13: RMSE Performance of Joint Estimation of Nonlinear Solution for M=3, σn = 60m and 103 Monte Carlo trials

We take 𝜎𝑛 = 30𝑚 and M=8 in Figure 14. The estimate is better as expected.

Figure 14: RMSE Performance of Joint Estimation of Nonlinear Solution for M=8, σn = 30m and 103 Monte Carlo trials

0 1 2 3 4 5 6 7 8 9 1068

70

72

74

76

78

80

82

84

86

88

RM

SE

(m

)

e,FDOA

(Hz) or e,TDOA

(x10-10

sec)

0 1 2 3 4 5 6 7 8 9 1010

20

30

40

50

60

70

80

90

RM

SE

(m

)

e,FDOA

(Hz) or e,TDOA

(x10-10

sec)



Conclusion

In this report TDOA, FDOA and joint TDOA/FDOA localization methods are discussed. Equations

are computed for estimating target position and corresponding simulations are done in MATLAB

environment. In the simulations, RMSE performance of each method is plotted versus

measurement error. In TDOA method, it is observed that linear solution outperforms Taylor

expansion approach. In FDOA simulations, nonlinear equations are computed with Matlab’s

specific function named fsolve. Estimation process is more computationally complex compared

to TDOA methods but the estimates are very well. In the last part, simulation results of joint

TDOA/FDOA method is shown. Less number of sensors are used compared to TDOA and FDOA

methods. It is observed that joint method gives two times more equations and it finds good

estimates under limiting conditions. Lastly, it is seen that in all methods, using more sensors and

making small errors yield perfect target estimation.

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